Questions on the usage and meaning of words in mathematics, the names for mathematical entities, and other such questions.

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2
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4answers
99 views

Origin of the terminology “Completeness” in R

When we study Real Analysis first, we usually analyse R equipped with the usual metric space. We intuitively make a picture of R as a line on which all the points are laid. We say that the set Q is ...
6
votes
2answers
58 views

Can the “generalized commutator bracket” $[(a,b,c)]$ be written using only the usual commutator bracket and $\mathbb{Q}$-linear combinations?

Let $\mathbb{K}$ denote a field. Background conventions. All my $\mathbb{K}$-algebras are associative. Given a finite set $X$, write $E(X)$ for the set of even permutations of $X$, and $O(X)$ for ...
3
votes
1answer
75 views

Is there a concept already for $\frac{f'(x)}{f(x)/x}$?

Given a differentiable real function $y=f(x)$, is there a math concept/terminology already defined for $$\frac{f'(x)}{f(x)/x}?$$ This quantity is inspired from price elasticity of demand. Thanks.
2
votes
2answers
22 views

The number of logarithm applications to get from n below 1

Let $L(n)$ to be a number of logarithms that you need to apply on $n$ until you get below 1: $$ 0 \leq \log\cdots\log n < 1 \\ \uparrow \\ L(n)\mbox{-times} $$ Is there a name for this function? ...
0
votes
0answers
27 views

Confusion regarding a statement in Atiyah-Macdonald

Atiyah-Macdonald says the following: If the ideals $a_i, a_j $ are co prime, then $\Pi a_i= \cap a_i $ What does this even mean? For example, we know that $(2), (3) $ are co prime in the ring ...
3
votes
4answers
119 views

Is there a purpose behind a function?

As I understand it, a function is a relation between two sets of numbers where as for every input value there is only assigned one output. Or for every $x$ there is only one $y$. What I don't ...
1
vote
2answers
44 views

Is “angle between two directions” appropriate?

I know vectors have both a magnitude and a direction, and I know that one may calculate the angle between two vectors. I am reviewing an academic paper where one of the author has written " This is ...
5
votes
3answers
1k views

How can the axiom of choice be called “axiom” if it is false in Cohen's model?

From what I know, Cohen constructed a model that satisfies $ ZF\neg C $. But if such a model exists, how can AC be an axiom? Wouldn't it be a contradiction to the existence of the model? Only ...
2
votes
1answer
41 views

Name for continuous maps satisfying $\operatorname{cl}(f^{-1}f(U))= \operatorname{cl}(U)$

I have recently come across particularly kind of continuous maps $f \colon X \to Y$ between topological spaces with the property that $$ \operatorname{cl}(f^{-1}f(U))= \operatorname{cl}(U), $$ for ...
2
votes
0answers
65 views

Is there a name for this property among variables?

I have a convex function of four variables, $f(w,x,y,z)$, which when solving for the symbolic arg min of one variable assuming the other three are known I got something similar to the following. ...
2
votes
1answer
28 views

Absolute continuity counterexample of a stochastic process

This example is from Stochastic Modelling and Applied Probability by Sören Asmussen (2010) p.358. The setup is the following: Let $\{Z_{t}\}$ be stochastic process on a Skorokhod space $D$ and a ...
3
votes
0answers
43 views

Is there a name for functions $f$ such that $\{f \le a\}$ is analytic?

Let $X$ be a Polish space and $f : X \to \mathbb{R}$ an arbitrary function. In The limit inferior of Borel functions I showed that a certain function $f$ has the following property: For each $a ...
0
votes
3answers
48 views

What's logical symbol for “for some”?

"For every" $x\in S$ would be $\forall x\in S$ which it's same as "for all" $x\in S$. But, is "for some" is same as "there exist"? It seems Yes, but is it Yes for every time? In several texts I found ...
2
votes
2answers
72 views

Do “small” and “large” numbers actually exist in an absolute sense?

Numbers like $(10)^{-10^{10^{10}}}$ are generally regarded as small, whereas numbers like, for example, Graham's Number, are regarded as extremely large. My question is, are these numbers simply ...
3
votes
0answers
24 views

On a metric over m-subsets of [n]

Given an integer $n$, denote the set of integers $\{1,2,\dots,n\}$ as $[n]$. For two $m$-subsets $A$ and $B$ of $[n]$, list their elements in the increasing order as $a_1 < a_2 < \dots < a_m$ ...
10
votes
1answer
531 views

The word “integral” in calculus unrelated to “integral” / “integer” in algebra?

I think that the word integral in calculus is nothing to do with integer or integer numbers. But why is integral is chosen for integration? In algebra, integral means related to integers, and this is ...
1
vote
0answers
21 views

Directed Acyclic Graph - root and leaf node terminology

I have found conflicting terminology regarding how to label nodes in directed acyclic graphs. Specifically, I am looking for a definition of root and leaf nodes (preferably something to cite). For ...
2
votes
1answer
39 views

How to read $[G:N]$?

For a group $G$ and a normal subgroup $N$ of $G$, the quotient group of $N$ in $G$ is written $G/N$. I could find from this link how to read $G/N$ ("$G$ modulo $N$" or "$G$ mod $N$"), but I couldn't ...
3
votes
1answer
54 views

What is the precise difference between functions and operators?

I have heard affirmatively that all operators are functions, but not all functions are operators. But at the same time I have heard that functions map numbers to numbers, whereas operators map ...
0
votes
0answers
49 views

Confusion regarding continuous functions between topological spaces – a subtle but possibly important point

Let $T: V_1 \to V_2$ be a linear mapping. Show that $T$ is a continuous function between $(V_1, \tau_{V_1}) $ and $(V_2, \tau_{V_2}) $ A direct solution to the problem is not what I am looking ...
4
votes
0answers
47 views

Terminology in graph theory

Let $G$ be a finite graph with the following property: For any vertex $a$ and edge $\{b, c\}$ of $G$, there is an edge connecting them: there is one of $\{a,b\}$ or $\{a, c\}$ in $G$. Is there ...
1
vote
1answer
30 views

What do you call a graph where the vertices are signed?

Let $G = (V, E)$ be a graph, and $f: V \to \{1,-1\}$ be a function assigning a sign to each vertex. What is this system $(G, f)$ called? In my current research, we've been using "oriented graph" ...
0
votes
0answers
6 views

Successive divisibility of a sequence? Progressive divisibility? terminology or reference

Perhaps I say that an (infinite) sequence $(r_n)$ of positive integers is progressively divisible iff $r_n \mid r_{n+1}$ for all $n$. Is there some other terminology that is in use for this? I am ...
-2
votes
1answer
28 views

The difference between descriptive and inferential statistics [closed]

What is the difference between descriptive and inferential statistics? How to identify from a sentence the difference of the two?
1
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1answer
13 views

Name for a collection of subsets of a set $E$ such that every element of $E$ is in some member?

Let $X$ be a set. A partition of $X$ is a collection of subsets $\{E_\alpha\}\subset\mathcal{P}(X)$ such that the following holds. For every $x\in X$, there is some $\alpha$ such that $x\in ...
1
vote
0answers
40 views

Vector calculus vs Vector analysis?

I was just wondering, is vector analysis the same as vector calculus? What about multivariable calculus? Because my multivariable calculus book (which I assume is the same as vector calculus?) covers ...
2
votes
1answer
34 views

Is there a name for the vector field that points along contour lines of a scalar field, proportional to the gradient?

Given a scalar field $G$ on $\mathbb{R}^2$ (say), the vector field $(\frac{\partial G}{\partial x}, \frac{\partial G}{\partial y})$ is called the gradient of $G$. Is there a standard name for the ...
0
votes
1answer
73 views

Terminological conventions regarding group actions

Suppose: $G$ is a group $T$ is a Lawvere theory ("algebraic theory") and $X$ is a $T$-algebra What conventions surround the phrase: "action of $G$ on $X$"? In particular, does this mean: a ...
0
votes
0answers
21 views

The basis induced by the nilpotent linear transformation

In Halmos's book, there is a theorem regarding the nilpotent transformation: If $A$ is a nilpotent linear transformation of index $q$ on a finite-dimensional vector space $V$, then there exist ...
0
votes
1answer
35 views

What does “421 is the smallest prime formed by the powers of two in logical order from right to left” mean and if so is it correct?

I've seen this on number gossip and a few other places, but I'm not exactly sure what it means. The only possibilities I have thought of for what they mean are "421 is the smallest center squared ...
3
votes
2answers
33 views

What is the name for one side of a ratio?

Basic example: "If you are asked to put a ratio in the simplest form, make sure that you have found the smallest factor that goes into both [?]." I've tried searching for ratio diagrams in Google, ...
0
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0answers
65 views

What is a nontrivial graph?

I've been operating happily under the definition that a nontrivial graph is a graph with at least two vertices for some time. Today I came upon a source which defined a nontrivial graph as a graph ...
1
vote
0answers
25 views

What to call a polynomial that with no roots in $\mathbb{Q}$ but does in $p$-adics

As an example, consider the polynomial $f(x)=x^2-2$. This polynomial has no roots over the rationals $\mathbb{Q}$. However, $\sqrt{2} \in \mathbb{Q}_7$, so it does have roots in the 7-adics. Is there ...
3
votes
1answer
32 views

Name for a nonlinear version of bilinear form

A map $b:X \times Y \to \mathbb{R}$ is called a bilinear form if $b$ is linear in both arguments. Is there a name for a form $b$ which is linear in only one argument and may be nonlinear in the ...
2
votes
0answers
40 views

What does it mean when two sets are “adjoined” in a metric space?

I encountered the word "adjoined" in Baby Rudin, Chapter 2 concerning basic topology on Euclidean space. It appeared in the proof to Theorem 2.35 Theorem$\quad$ Closed subsets of compact sets are ...
0
votes
0answers
17 views

What does it mean an ideal is nilpotent modulo another ideal?

Reference:Chatters, A. W.; Hajarnavis, C. R. (1971), "Non-commutative rings with primary decomposition", Quart. J. Math. Oxford Ser. (2) 22: 73–83, doi:10.1093/qmath/22.1.73 Let $R$ be an rng and ...
1
vote
1answer
24 views

Terminology help for a set relation: for sets $X, Y$, not necessarily disjoint, such that neither is a subset of the other.

Is there an existing term for pairs of sets $X, Y$, not necessarily disjoint, such that neither $X \subseteq Y$ nor $Y \subseteq X$? Would it be incorrect (or misleading) to call them something like ...
1
vote
1answer
21 views

Definition of quasi-cyclic and full rational groups

In Unit Groups of Classical Rings by Karpilovsky, p.96, we can see this theorem: Let $G$ be a divisible abelian group. Then $G$ is a direct product quasi-cyclic and full rational groups. I want ...
1
vote
1answer
43 views

Name when two functions are equal under integration (expectation)?

What is it called when $E[X] = E[Y]$? That is, $$\int x f(x)\,dx = \int y g(y)\,dy.$$ What I want to say is not that the expectation of $X$ is equal to that of $Y$ but rather (the equivalent ...
4
votes
2answers
135 views

What do we call the covector associated to a vector?

Let $V$ denote an inner product space. Write $V^*$ for either the algebraic dual, or else the continuous dual. In either case, for each vector $v \in V$, we get a covector $v^c \in V^*$ given by: ...
4
votes
0answers
113 views

When do the zero divisors of a commutative ring form an ideal?

Let $J$ denote the set of zero-divisors of a commutative ring $R$. Since we automatically have $RJ \subseteq J$, hence $J$ is automatically halfway to being an ideal. Furthermore, its already ...
2
votes
0answers
40 views

Proper names for different representations of the same formula

I would like to know what to call formulas that are all on one line and what to call the same formulas that are on multiple lines. One line example: P ÷ TVD ÷ 0.052 Multiple line example: ...
3
votes
1answer
29 views

Does “data” in Cauchy data come before or after the coinage of data in computer science

Is the usage of data as in Cauchy data (i.e. initial conditions) borrowed or came before the usage of data in computer science and do both usages mean roughly the same thing (data ~ information)?
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0answers
14 views

Is there a term for irreflexive relations whose reflexive closures are equivalence relations?

Consider a relation $\sim$ which is irreflexive, that is: $$ \forall A \:A\nsim A $$ Further suppose the reflexive closure of $\sim$ is an equivalence relation. The reflexive closure of $\sim$, ...
1
vote
1answer
47 views

prior probability vs a priori probability

What is the difference between "Prior probability" and "a priori probability" Wikipedia have two distinct pages for them. As of my inference i thought "Prior" and "a priori" are same, i.e., P(y) in ...
3
votes
0answers
25 views

By or through for a rotation

"Rotated through pi rad" vs "Rotated by pi rad" I have heard both used and also heard mentioned that there was a mathematical difference between the two. Is this true or can they be used ...
7
votes
5answers
416 views

Necessity of being Hausdorff in the definition of compactness?

According to R Engelking - General Topology: A topological space $X$ is called a compact space if $X$ is a Hausdorff space and every open cover of $X$ has a finite subcover, i.e., if for every ...
0
votes
0answers
31 views

Name (reference?) for lax monoidal functors that are 'full, or surjective, on monoids'?

A lax monoidal functor $F$ takes monoids to monoids. Is there a name for a lax monoidal functor that is 'full' or surjective with respect to this property? In other words, the functor $F : \mathcal{C} ...
2
votes
3answers
96 views

Is there any difference between 'all real numbers' and '$(-\infty, \infty)$'

I've just thought about this. All the textbooks I've been looking at for pre-calc, the domains are always written as 'all real numbers', whereas my calculus textbooks would rather write them as ...
0
votes
0answers
27 views

Name of a vertex set of the same out-degree

I have a graph and it is very important to me to distinguish vertex sets of the same out-degree. For example, I have a set of all vertex of the out-degree 1, a set of all vertex of out-degree2, and so ...